Stochastic Analysis of Life Insurance Surplus

Transcription

1 Stochastic Analysis of Life Insurance Surplus Natalia Lysenko, Gary Parker Abstract The behaviour of insurance surplus over tie for a portfolio of hoogeneous life policies in an environent of stochastic ortality and rates of return is exained. We distinguish between stochastic and accounting surpluses and derive their first two oents. A recursive forula is proposed for calculating the distribution function of the accounting surplus. We then exaine the probability that the accounting surplus becoes negative in a given insurance year. Nuerical exaples illustrate the results for portfolios of teporary and endowent life policies assuing a conditional AR1 process for the rates of return. Keywords: insurance surplus, stochastic rates of return, AR1 process, stochastic ortality, distribution function. 1 Introduction The surplus is an iportant indicator of an insurance copany s financial position. In this paper, we present a ethodology for studying the insurance surplus for a hoogeneous portfolio of life insurance policies in a stochastic ortality and rates of return environent. The results are presented in a rather siplified fraework with the ain focus on the cash flows arising fro just the benefit and preiu payent streas driven by the ortality experience of the portfolio; i.e. expenses and other possible sources of decreents e.g. lapses are ignored. A siilar approach for the investent coponent as in Parker 1997 is assued. That is, a rando global rate of return is used to deterine values of the cash flows. Although there are any liitations of our fraework to ore fully address coplexity of the real world, the results and conclusions are believed to be useful in enhancing actuaries understanding of the stochastic behaviour underlying life insurance products. The study of life contingencies in an environent of stochastic ortality and interest rates can be traced back to the 1970 s, and by now there exists a vast actuarial literature on the topic. In particular, soe of the papers that consider portfolios of life insurance and life annuity contracts include Frees 1990, Parker 1994a, 1996, 1997 and Coppola et al Marceau and Gaillardetz 1999 look at the reserve calculation for general portfolios of life insurance policies and exaine the appropriateness of using the liiting portfolio approxiation. For an extensive literature review on the subject, the reader is referred to Hoedeakers et al. 2005; the paper proposes an approxiation for the distribution of the prospective loss for a hoogeneous portfolio of life annuities based on the concept of coonotonicity. In general, the above-entioned papers deal with a stochastically discounted value of future contingent cash flows that are viewed and valued at the sae point in tie. This includes net single preiu and reserve calculations. Dept. of Statistics & Actuarial Science, Sion Fraser University, BC, Canada. 1

2 The proble we try to address is of a different nature. To illustrate, consider a closed block of life insurance business at its initiation, tie 0. At each future valuation date, we are interested in the financial position of this block of business as easured by the aount of the surplus available at that tie. Let us fix one of the future valuation dates, say corresponding to tie r, and consider how the surplus can be described at this valuation date. Prior to tie r, the insurer collects preius and pays death benefits according to the ters of the contracts in the portfolio. The accuulated value to tie r of these cash flows represents the insurer s retrospective gain or the assets accuulated fro this block of business. After tie r, the insurer continues? to pay benefits as they coe due and receive periodic preius. This strea of payents, discounted to tie r, constitutes the prospective loss, which represents net future obligations or liabilities of the insurer. This leads to one definition of the surplus as the difference between the retrospective gain and the prospective loss. We also consider an alternative definition of the surplus referred to as the accounting surplus with the liabilities given by the actuarial reserve as opposed to the prospective loss itself. One can draw an analogy between our approach and the dynaic solvency testing DST, which involves projecting a copany s solvency position into the future under varying assuptions fro 1994 General Insurance Convention, by J. Charles. However, instead of considering several scenarios, we average over all possible scenarios by placing a distribution on the. The first three sections set up our fraework for studying hoogeneous portfolios assuing rando ortality and rates of return. In Sections 5, 6 and 7 we define the two types of surplus and derive their first two oents. For insurance regulators it is iportant that insurance copanies aintain an adequate surplus level. To represent actuarial liabilities, the insurers are required to report their actuarial reserves calculated in accordance with the regulations. So, when onitoring insurance copanies, the regulators actually look at what we call the accounting surplus. In Section 8, we propose a forula for obtaining the distribution function of the accounting surplus at a given valuation date. One piece of inforation that is readily available fro this distribution function is the probability that the surplus falls below zero. If this probability is too high, say above 5%, then perhaps the insurer should ake soe adjustents to the ters of the contract such as, for exaple, increasing the preiu rate or raising additional initial surplus.in Section 8, a ethod for coputing the distribution function of the accounting surplus is discussed. Finally, in Section 9, nuerical exaples for two portfolios of endowent and teporary life insurance policies are used to illustrate ain results of this paper. 2 Hoogeneous Portfolio In this paper, we consider a portfolio of identical life policies issued to a group of policyholders all aged x with the sae risk characteristics. It is assued that the future lifeties of the policyholders in the portfolio are independent and identically distributed. Each policy pays a death benefit b at the end of the year of death if death occurs within n years since the policy issue date and a pure endowent benefit c if the policyholder survives to the end of year n. The annual level preiu π is payable at the beginning of each year as long as the contract reains in force. 2

3 3 Cash Flows To study the surplus for a portfolio, one approach would be to odel the surplus for a single policy and then su over the individual policies in the portfolio. However, this approach can be coputationally tie-consuing when dealing with large portfolios. Alternatively, one could odel aggregate annual cash flows and su over policy-years. This is the way we will proceed. Consider a valuation date corresponding to tie r. For the purpose of our further discussion, we distinguish between those cash flows that occur prior to tie r and those that occur after. In addition, we define the cash flows prior to tie r as the net inflows into the copany and the cash flows after tie r as the net outflows. More forally, let RCj r denote the net cash inflow at tie j for 0 j r i.e., it is the retrospective cash inflow for the valuation at tie r. It is given by ] RCj r = π L i,j 1 {j<r} b D i,j 1 {j>0} i=1 = π i=1 L i,j 1 {j<r} b i=1 D i,j 1 {j>0} = π L j 1 {j<r} b D j 1 {j>0}, 3.1 where { 1 if policyholder i is alive at tie j, L i,j = 0 otherwise, { 1 if policyholder i dies in year j, D i,j = 0 otherwise, and 1 {A} is an indicator function; it is equal to 1 if condition A is true and 0 otherwise. Notice that at j = 0, RC0 r is siply the su of all the preius collected at the issue date. Now consider what happens at the valuation date, j = r. Death benefits are paid at the end of year r to everyone who dies during that year. So, this cash flow becoes a part of the retrospective cash inflow RCr r. However, preius are collected at the beginning of the next year r + 1 st year and, therefore, they contribute to the prospective cash outflow defined below. L j = i=1 L i,j is the nuber of people fro the initial group of policyholders who survive to tie j i.e. it is the nuber of in-force policies at tie j and D j = i=1 D i,j is the nuber of deaths in year j. Let P Cj r denote the net cash outflow that occurs j tie units after tie r for 0 j i.e., it is the prospective cash outflow for the valuation at tie r. Using notation introduced above, we have ] P Cj r = b D i,r+j 1 {j>0} + c L i,n 1 {j=} π L i,r+j 1 {j<} i=1 = b D r+j 1 {j>0} + c L n 1 {j=} π L r+j 1 {j<}. 3.2 Calculation of the cash flow oents is straightforward. For exaple, ERC r j ] = π EL j ] 1 {j<r} b ED j ] 1 {j>0} ; 3

4 VarRC r j ] = π 2 VarL j 1 {j<r} + b 2 VarD j 1 {j>0} 2 π b CovL j, D j 1 {0<j<r} and, for i < j, CovRC r i, RC r j ] = π 2 CovL i, L j ] 1 {j<r} + b 2 CovD i, D j ] 1 {i>0} π b CovL i, D j ] 1 {0<i<j<r} b π CovD i, L j ] 1 {0 i<j r}. Using the fact that, under our assuption of the hoogeneous portfolio and independent future lifeties 1, {D j } r j=1 {L r} is ultinoial ; 0 q x, 1 q x,..., r 1 q x, r p x, the oents of D j and L j can be calculated. For instance, refer to Equations 9 and 10 in Marceau and Gaillardetz 1999 for explicit expressions with r i = 0, since we study all cash flows as viewed fro tie 0. 4 Stochastic Rates of Return For illustrative purposes, we choose to odel the forces of interest by a conditional autoregressive process of order one, AR1, given the current force of interest. This odel was used by Bellhouse and Panjer 1981 and Marceau and Gaillardetz 1999,for instance. Let δk be the force of interest in period k 1, k], k = 1, 2,..., n, with a possible realization denoted by δ k. Then, the forces of interest satisfy the following autoregressive odel: δk δ = φ δk 1 δ] + σεk, 4.1 where {εk} is a sequence of independent and identically distributed standard noral rando variables and δ is the long-ter ean of the process. We assue that φ < 1 to ensure covariance stationarity of the process. For our further discussion, it is convenient to introduce a notation for the force of interest accuulation function, which is then used to study both discounting and accuulation processes. Let Is, r denote the force of interest accuulation function between ties s and r, 0 s r 2. It is given by Is, r = { r j=s+1 δj, s < r, 0, s = r. 4.2 The AR1 odel has already been extensively studied and any results about it are readily available in the literature. Following siilar derivations as given, for exaple, in Cairns and Parker 1997, we get and EIs, r δ0 = δ 0 ] = r sδ + VarIs, r δ0] = r s + = σ2 1 φ 2 2φ r s 1 1 φ φ 1 φ φs φ r δ 0 δ 4.3 φ 1 φ 1 φr s 1 ] φ 1 φ 2 φ s φ r If K x denotes the cutate-future lifetie of a person aged x, then in the standard actuarial notation PK x = k = k 1 q x and PK x > k = k p x. 2 This notation is an extension of the notation introduced in Marceau and Gaillardetz

5 When two force of interest accuulation functions are involved, it will be necessary to distinguish three cases for the ties between which the accuulation occurs. Suppose we are interested in obtaining the value at tie r of two cash flows occurring at ties s and t. 1. If s < t < r i.e., both cash flows occur prior to tie r, the values at tie r of these cash flows need to be accuulated using Is, r and It, r; 2. If r < s < t i.e., both cash flows occur after tie r, the values at tie r of these cash flows need to be discounted using Ir, s and Ir, t; 3. If s < r < t i.e., one cash flow occurs before tie r and the other one occurs after tie r, the values at tie r of these cash flows need to be accuulated and discounted using Is, r and Ir, t respectively. Case 1 is considered in Cairns and Parker 1997, whereas Case 2 is a slight generalization of results presented in Marceau and Gaillardetz To suarize, the unconditional covariance ters, corresponding to the three cases for the force of interest accuulation functions entioned above, are given by the following forulas: Case 1: s < t < r CovIs, r, It, r] = Case 2: r < s < t r r i=s+1 j=t+1 CovIr, s, Ir, t] = VarIr, s] + Case 3: s < r < t CovIs, r, Ir, t] = Covδi, δj] = VarIt, r] + σ2 1 φ 2 φ 1 φ 2 φt φ r φ t φ s ; σ2 φ 1 φ 2 1 φ 2 φs φ t φ s φ r ; σ2 φ 1 φ 2 1 φ 2 φr φ t φ r φ s. The corresponding conditional covariance ters can then be obtained using known results fro the ultivariate noral theory e.g., see Johnson and Wichern For instance, for s < r < t Case 3, where CovIs, r, Ir, t δ0] = CovIs, r, Ir, t] CovIs, r, δ0] = Forulas for the other two cases are analogous. CovIs, r, δ0] CovIr, t, δ0], 4.5 Varδ0] σ2 φ 1 φ 2 1 φ φs φ r. 4.6 We will also require the distribution of Ir, t for r < t conditional on both δ0 and δr. In this case, due to the Markovian nature of the rates of return process, we have EIr, t δ0 = δ 0, δr = δ r ] = EI0, t r δ0 = δ r ] 5

6 and VarIr, t δ0, δr] = VarI0, t r δ0], and Equations 4.3 and 4.4 can be applied. To siplify notation, we will oit conditioning on δ0 when referring the oents of a function of Is, r e.g., E e Is,r δ0 ] will be denoted by E e Is,r]. In our notation, the accuulation function fro tie s to tie r and the discount function fro tie t to tie r, for s < r < t, are given by e Is,r and e Ir,t, respectively. Each of the follows a lognoral distribution. Using a well-known fact that if Y N EY ], VarY ], then the th -oent of e Y E e Y ] 2 EY ]+ = e 2 VarY ], 4.7 we can find oents of e Ir,t and e Is,r. Next, we introduce two rando variables, the retrospective gain and the prospective loss, which will be used to define and study the surplus. is 5 Retrospective Gain The retrospective gain at tie r is the difference between the accuulated values to tie r of past preius collected and benefits paid. It can be expressed in ters of RCj r as follows: RG r = r RCj r e Ij,r. 5.1 j=0 Then, assuing independence between future lifeties and rates of return, we obtain ERG r ] = r ERCj r ] Ee Ij,r ] 5.2 j=0 and RGr ] 2 E = r i=0 j=0 r ERCi r RCj r ] Ee Ii,r+Ij,r ] Prospective Loss The prospective loss at tie r is the difference between the discounted values to tie r as viewed fro tie 0 of future benefits to be paid and preius to be collected. Using P Cj r, it is given by P L r = P Cj r e Ir,r+j. 6.1 j=0 The oents of P L r can be calculated siilarly to the oents of RG r. 6

7 When calculating actuarial reserves, it will be useful to know the oents of the prospective loss conditional on the nuber of in-force policies and the rate of return prevailing at the valuation date. For exaple, we will need the conditional expectation of P L r. Due to the independence of ortality and interest, the following holds: EP L r L r, δr] = EP Cj r L r ] Ee Ir,r+j δr]. 6.2 j=0 To calculate the conditional expected value of P Cj r, the following facts can be used: and {L r+j L r = r } binoial r, j p x+r 6.3 {D r+j L r = r } binoial r, j 1 q x+r Surplus In general, we define insurance surplus to be the difference between assets and liabilities at a given valuation date. Recall that the retrospective gain is the accuulated value of past preius collected net of past benefits paid and, thus, in our context, it can be interpreted as the value of assets. In turn, the liabilities associated with a portfolio of life policies are based on the prospective loss, which is the discounted value of future obligations net of future preius. So, the liabilities can siply be represented by the prospective loss rando variable. In this case the surplus is referred to as the net stochastic surplus or just stochastic surplus and is denoted Sr stoch. S stoch r = RG r P L r. 7.1 In practice, at each valuation date, an insurer is required to set aside an actuarial reserve based on the current situation including the nuber of in-force policies as well as the prevailing rate of return. This reserve is a liability ite on the balance sheet of the insurance copany. So, an alternative definition of the surplus is the difference between the value of assets and the actuarial reserve, in which case we call it the accounting surplus, denoted S acct r. S acct r = RG r r V L r, δr, 7.2 where r V L r, δr is the actuarial reserve for the valuation at tie r. Although the nae ay suggest a deterinistic nature of the accounting surplus, in fact, it is a stochastic quantity for r > 0, since, when viewed fro tie 0, there is an uncertainty about both the nuber of in-force policies and the force of interest at tie r. The reserve is intended to cover the net future liabilities of the insurer. Therefore, the aount needed to be set as a reserve at tie r should be at least the expected value of P L r conditional on the nuber of in-force policies in the portfolio L r and the force of interest δr. If, instead, it is required to have a conservative reserve that will cover future obligations with a high probability, one can use a p th percentile of the prospective loss rando variable with p between 70% and 95%, for exaple. However, this reserve calculation can be fairly difficult to incorporate into the odel, since we need to know the distribution function of P L r, which is not 7

8 easy to obtain. Alternatively, a reserve could be set equal to the expected value plus a ultiple of the standard deviation of P L r as suggested in Norberg In the rest of this section, assue that rv L r, δr = E P L r L r, δr ]. 7.3 It is easy to show that under our particular choice of the reserve E Sr acct ] ] ] ] = E S stoch r = E RGr E P Lr. 7.4 The variance of the accounting surplus can be calculated using the following result. Result 7.1. where VarS acct r ] = Var δr EP L r δr] + E δr Var Lr EP Lr L r, δr] + VarRG r ] 2CovRG r, P L r, Var δr EP L r δr] + E δr Var Lr EP Lr L r, δr] = { = E Lr EP Ci r L r ] EP Cj r L r ] i=0 j=0 E δr E e Ir,r+i δr ] E e Ir,r+j δr ]} 2 EP L r ] and Cov ] ] ] ] RG r, P L r = E RGr P L r E RGr E P Lr r = ERCj r P Ci r ] Ee Ij,r Ir,r+i ] ERG r ] EP L r ]. j=0 i=0 A proof of Result 7.1 is given in Appendix A. The variance calculation for the stochastic surplus is straightforward as Var Sr stoch ] ] = Var RGr P L r = Var ] ] ] RG r + Var P Lr 2 Cov RGr, P L r. We also consider a portfolio with the nuber of policies approaching infinity, which we refer to as the liiting portfolio. Although the liiting portfolio is an abstract concept and not achievable in practice, its characteristics such as variability can serve as bencharks for portfolios of finite sizes and can provide soe useful inforation for insurance risk anagers. If the variance of the surplus per policy for a given portfolio is uch larger than the corresponding variance for the liiting portfolio, then it can be concluded that a large portion of the total risk is due to the insurance risk. In other words, there is a great uncertainty about future cash flows. One iplication of this is that, if the insurer decides to hedge the financial risk, for instance, by purchasing bonds whose cash flows will atch those of the portfolio s liabilities, this strategy will not be very efficient and the cost incurred to ipleent it ight not 8

9 be justified. In this case, selling ore policies, sharing the ortality risk or buying reinsurance are better strategies to itigate the risk. For a liiting portfolio, the calculation of the oents is done siilarly to the case when the size of the portfolio is finite, except that the rando cash flows per policy, RCj r / and P Ci r /, are replaced by their expected values. 8 Distribution of Accounting Surplus In this section, we describe a ethod for calculating the distribution function of the accounting surplus. 8.1 Portfolio of finite size Recall that the accounting surplus at tie r is defined as S acct r = RG r r V L r, δr, where r V L r, δr is the reserve at tie r. Notice that, given the values of L r and δr, r V L r, δr is constant. Therefore, we can obtain the distribution function df of {Sr acct L r, δr} fro the df of {RG r L r, δr} using PS acct r ξ L r = r, δr = δ r ] = PRG r ξ + r V r, δ r L r = r, δr = δ r ]. Since it is not straightforward to get the distribution function of {RG r L r, δr} directly, we propose a recursive approach. This approach is based on the ideas presented in Parker1998. For the valuation at tie r, let G t = t j=0 RCr j eij,t denote the accuulated value to tie t of the retrospective cash inflows that occurred up to and including tie t, 0 t r. Observe that G r is equal to RG r. We can relate G t and G t 1 as follows: G t = t RCj r e Ij,t j=0 t 1 = RCj r e Ij,t 1+It 1,t + RCt r e It,t = j=0 t 1 RCj r e Ij,t 1 e δt + RCt r = G t 1 e δt + RC r t. 8.1 Equation 8.1 can be used to build up the df of G t fro the df of G t 1 and thus the df of RG r recursively fro G t for t = 0, 1,..., r 1. Note that 3 3 f denotes the probability density function pdf. Under our assuption for the rates of return, f δt is the pdf of a noral rando variable with ean Eδt δ0 = δ 0] and variance Varδt δ0 = δ 0], and f δt A, where A {δt 1 = δ t 1}, is the pdf of a noral rando variable with ean Eδt δ0 = δ 0, A ] and variance Varδt δ0 = δ 0, A ]. j=0 9

10 PG t λ L t = t, δt = δ t ] = PL t = t, δt = δ t G t λ] PG t λ] PL t = t, δt = δ t ] = PL t = t G t λ] f δt δ t G t λ PG t λ], 8.2 PL t = t ] f δt δ t where the last line follows fro the independence of L t and δt. Next we consider a function g t λ, t, δ t given by g t λ, t, δ t = PG t λ L t = t, δt = δ t ] PL t = t ] f δt δ t 8.3 and otivated by Equation 8.2. The following result gives a way for calculating g t fro g t 1, 1 < t r n. Result g t λ, t, δ t = t 1 = t P ] L t = t Lt 1 = t 1 λ ηt g t 1, t 1, δ t 1 f e δt δ t δt 1 = δ t 1 dδ t 1, δt where η t is the realization of RCt r for given values of t 1 and t, { π t b η t = t 1 t, 1 t r 1, b t 1 t, t = r, with the starting value for g t { PL1 = g 1 λ, 1, δ 1 = 1 ] f δ1 δ 1 if G 1 λ, 0 otherwise. Proof: Fro Equation 8.2 we have g t λ, t, δ t = PL t = t, δt = δ t G t λ] PG t λ] = PL t 1 = t 1, L t = t, δt = δ t G t λ] PG t λ] t 1 = t = PG t λ L t 1 = t 1, L t = t, δt = δ t ] t 1 = t PL t 1 = t 1, L t = t, δt = δ t ] by the Bayes rule. Using Equation 8.1, which iplies that { } { } G t λ G t 1 λ RCr t, and the assuption e δt 10

11 of independence of L t 1 and L t fro δt, we get g t λ, t, δ t = = t 1 = t P t 1 = t P G t 1 λ η ] t Lt 1 = t 1, L t = t, δt = δ t e δt PL t 1 = t 1, L t = t ] f δt δ t L t 1 = t 1, L t = t Gt 1 λ η ] t f e δt δ t Gt 1 λ η t δt e δt P G t 1 λ η ] t e δt = P L t = t Lt 1 = t 1, G t 1 λ η ] t e δt t 1 = t P L t 1 = t 1 Gt 1 λ η ] t P G t 1 λ η ] t e δt e δt δt f δt δ t 1 = δt 1, G t 1 λ η t f e δt 1 δ t 1 G t 1 λ η t dδ t 1. δt e δt By the Markovian property of L t and δt and the definition of g λ ηt t 1, e δ t t 1, δ t 1, g t λ, t, δ t becoes g t λ, t, δ t = t 1 = t P ] L t = t Lt 1 = t 1 λ ηt g t 1, t 1, δ t 1 f e δt δ t δt 1 = δ t 1 dδ t 1. δt S acct r Once g r λ, r, δ r is obtained using Result 8.1.1, the cuulative distribution function of can be calculated as follows: PSr acct ξ] = r=0 PS acct r ξ L r = r, δr = δ r ] PL r = r ] f δr δ r dδ r. Since S acct r PS acct = = = RG r r V L r, δr and G r = RG r, we get r ξ] = PG r ξ + r V r, δ r L r = r, δr = δ r ] PL r = r ] f δr δ r dδ r r=0 r=0 g r ξ + r V r, δ r, r, δ r dδ r Liiting Portfolio For a very large insurance portfolio, the actual ortality experience follows very closely the life table. In this case, we can approxiate the true distribution of the surplus per policy by the distribution of the surplus for the liiting portfolio, which takes into account the investent risk but treats cash flows as given and equal to their expected values. 11

12 This distribution for the liiting portfolio can be derived siilarly to the case of rando cash flows. Define G t = t j=0 ERCr j /] eij,t. It can easily be shown that G t = G t 1 e δt +ERC r t /] cf. Equation 8.1. Now, let h t λ, δ t = PG t λ δt = δ t ] f δt δ t. This function can be used to calculate the df of G t recursively siilar to the way g t λ, t, δ t was used for obtaining the df of G t. A recursive relation for h t λ, δ t is given by the following result. Result h t λ, δ t = f δt δ t δt 1 = δ t 1 h t 1 λ ERC r t /] e δt, δ t 1 dδ t 1 with the starting value for h t { fδ1 δ h 1 λ, δ 1 = 1 if G 1 λ, 0 otherwise. Since li PS acct r / ξ δr = δ r ] = PG r ξ + r Vδr δr = δ r ], li PSacct r / ξ] = h r ξ + r Vδr, δ r dδ r, 8.5 where r Vδr denotes the benefit reserve at tie r per policy for the liiting portfolio. 9 Nuerical Illustrations We use the following arbitrarily chosen values of the paraeters for the conditional AR1 process: φ = 0.90; σ = 0.01; δ = 0.06 and δ 0 = A nonparaetric life table is used to deterine the distribution of K x. In our exaples we use the Canada 1991, age nearest birthday ANB, ale, aggregate, population ortality table; see Appendix for ortality rates. We also assue that the actuarial reserves are set equal to the conditional expected value of the prospective loss. That is, rv L r, δr = E P L r L r, δr ], and r V δr = E P L r / δr ]. In our exaples, we copare results for different preiu rates. A θ% preiu loading is applied to the benefit preiu deterined under the equivalence principle; i.e. the preiu such that EP L 0 ]=0 is satisfied. Here, a copany with a negative surplus in any year is considered insolvent. 12

13 9.1 Exaple 1: Portfolio of endowent policies Consider a portfolio of year endowent life insurance policies with $1000 death and endowent benefits issued to a group of people aged 30 with the sae risk characteristics. Tables 1 and 2 provide the first three oents of the accounting surplus per policy represented by the expected value, standard deviation and coefficient of skewness for the portfolio of 100 endowent contracts and the corresponding liiting portfolio, respectively. The estiates of the probability of insolvency based on the accounting surplus are given in Table 3. First of all, one can see fro both the oents and the probability estiates that the difference between the two portfolios is not very large. In this case, approxiation of the distribution of the surplus per policy for portfolios of oderate size by the distribution of the surplus per policy for the corresponding liiting portfolio ay be justified. An insignificant loss in the accuracy of the estiates is well copensated by a gain in coputing tie. When charging the benefit preiu θ = 0%, probabilities for all values of r are slightly less than 50%. This can be expected since no profit or contingency argin is built into the preiu when pricing is done under the equivalence principle. The fact that these probabilities are not exactly 50% is due to the asyetry of the discount function when a Gaussian odel for the rates of return is assued. With a 10% loading factor, the probability of insolvency sharply decreases copared to the case of θ = 0% in the first few years but this reduction is not as large in the later years of the contract. The probability that the accounting surplus falls below zero increases fro 0.23% at r = 1 to 14.57% at r = 10 for = 100. A 20% loading factor appears to be sufficient to ensure that the probability of insolvency in any given year is less than 5% for both portfolios. Table 1: The first three oents of the accounting surplus per policy for a portfolio of year endowent policies. θ = 0%, π=67.90 θ = 10%, π=74.69 θ = 20%, π=81.48 r E ] sd ] ŝk ] E ] sd ] ŝk ] E ] sd ] ŝk ]

14 Table 2: The first three oents of the accounting surplus per policy for the liiting portfolio of 10-year endowent policies. θ = 0%, π=67.90 θ = 10%, π=74.69 θ = 20%, π=81.48 r E ] sd ] ŝk ] E ] sd ] ŝk ] E ] sd ] ŝk ] Table 3: Estiates of the probability that accounting surplus per policy becoes negative for two portfolios of 10-year endowent policies. = 100 θ = 0% θ = 10% θ = 20% θ = 0% θ = 10% θ = 20% r π = π = π = π = π = π =

15 9.2 Exaple 2: Portfolio of teporary policies In our next exaple, we study a hoogeneous portfolio of year teporary insurance policies and the corresponding liiting portfolio with $1000 death benefit issued to people aged 30. In contrast to the portfolio of endowent policies, the oents of the accounting surplus per policy for the 1000-policy portfolio and the liiting portfolio of teporary contracts cf. Tables 4 and 5 are largely apart. This is confired by their distribution functions as well; see Table 6 for the estiates of the probabilities of insolvency. Preius with 2% or 3% loading factors considerably decrease the probability of insolvency over the whole ter of the contracts for the liiting portfolio but have essentially no ipact on those probabilities for the 1000-policy portfolio. Even a 20% loading factor is not sufficient to reduce the probabilities of insolvency to a reasonably low level e.g. 5-10%. An iplication of this is that for portfolios of teporary insurances, an insurer either has to aintain a very large portfolio or use a large preiu loading. The distribution of the surplus for the 1000-policy portfolio reains negatively skewed for all values of r. In the case of the liiting portfolio, skewness coefficients are fairly sall in agnitude and change fro being negative for sall values of r to being positive for larger r. In the case of the 1000-policy portfolio, note that the plots for sall values of r look ore like plots of step functions for the df of a discrete rando variable. This should not be a surprise. Reeber that the surplus depends on the two rando processes - a continuous one for the rates of return and a discrete one for the decreents. In the earlier years of the teporary contract, only a few deaths are likely to occur but each of the would have a relatively large ipact on the surplus. This is reflected in the jups of the df of Sr acct /. The slightly upward sloped segents of the plots between any two jups indicate very sall probabilities that the surplus realizes values in those regions. But in the later years the shape of the df gradually soothes out due to the fact that there are ore possibilities for allocating death events over the past years. Table 4: The first three oents of the accounting surplus per policy for a portfolio of year teporary policies. θ = 0%, π=1.27 θ = 3%, π=1.31 θ = 20%,π=1.52 r E ] sd ] ŝk ] E ] sd ] ŝk ] E ] sd ] ŝk ]

16 Table 5: The first three oents of the accounting surplus per policy for the liiting portfolio of 5-year teporary policies. θ = 0%, π=1.27 θ = 2%, π=1.29 θ = 3%, π=1.31 r E ] sd ] ŝk ] E ] sd ] ŝk ] E ] sd ] ŝk ] Table 6: Estiates of the probability that accounting surplus per policy becoes negative for two portfolios of 5-year teporary policies. = 1000 θ = 0% θ = 3% θ = 20% θ = 0% θ = 2% θ = 3% r π = 1.27 π = 1.31 π = 1.52 π = 1.27 π = 1.29 π =

17 10 Conclusions The paper exained the insurance surplus at different future valuation ties viewed fro the point of the contracts initiation. An advantage of this fraework is that it allows to assess adequacy of, for exaple, initial surplus level, pricing and reserving ethod before the block of business is launched, and so any necessary odifications to the ters of the contract can be ade. It was suggested to distinguish between two types of insurance surplus, naely stochastic and accounting surpluses, each serving a slightly different purpose in addressing insurer s solvency. The accounting surpluses represent the financial results as they will be seen by shareholders and regulators at future valuation dates. When studying the stochastic surplus, one considers the range of possible portfolio values easured at a given valuation date that could becoe reality once all contracts in the portfolio have atured. The distribution function of the accounting surplus was nuerically obtained by applying the proposed recursive forula. The analysis of the probabilities of insolvency based on the accounting surplus was used to coent on the adequacy of preiu rates. In fact, the probability of insolvency can be used as a risk easure. For exaple, fro the nuerical illustrations, we saw that the preiu loading required to ensure a sufficiently sall probability of insolvency is uch larger for a sall portfolio than it is for a very large portfolio in the case portfolios of teporary policies. The recursive forula for the distribution of the accounting surplus took advantage of conditionally constant liability. Obtaining the distribution of the stochastic surplus is a uch harder proble since one needs to take into account both rando assets and liabilities. Another interesting question would be to calculate the probability of solvency over a tie horizon, finite or infinite. The odel can be ade ore realistic by including expenses and lapses. 11 Acknowledgeents Financial support fro the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged. References 1] Bellhouse, D.R., and H.H. Panjer Stochastic odelling of interest rates with applications to life contingencies - Part II. Journal of Risk and Insurance, 484: ] Bowers, N.L., H.U. Gerber, J.C. Hickan, D.A. Jones, and C.J Nesbitt Actuarial Matheatics Second Edition. Society of Actuaries, Schauburg, Illinois. 3] Cairns, A.J.G., and G. Parker Stochastic pension fund odelling. Insurance: Matheatics and Econoics, 21: ] Coppola, M., E. Di Lorenzo and M. Sibillo Stochastic analysis in life office anageent: applications to large annuity portfolios. Applied Stochastic Models in Business and Industry, 19:

18 5] Frees, E.W Stochastic life contingencies with solvency considerations. Transactions of the Society of Actuaries, 42: ] Hoedeakers, T., G. Darkiewicz and M. Goovaerts Approxiations for life annuity contracts in a stochastic financial environent. Insurance: Matheatics and Econoics, 37: ] Johnson, R.A., and D.W. Wichern Applied Multivariate Statistical Analysis. 5th edn. Prentice-Hall, NJ. 8] Marceau, E., and P. Gaillardetz On life insurance reserves in a stochastic ortality and interest rates environent. Insurance: Matheatics and Econoics, 25: ] Melsa, J.L., and A.P. Sage An introduction to Probability and Stochastic Processes. Prentice-Hall, New Jersey. 10] Norberg, R A solvency study in life insurance. Proceedings of the third AFIR International Colloquiu, Roe, ] Pandit, S.M., and S. Wu Tie Series and Syste Analysis with Applications. John Wiley & Sons, New York, 586pp. 12] Parker, G. 1994a. Liiting distribution of the present value of a portfolio. ASTIN Bulletin, 241: ] Parker, G. 1994b. Two stochastic approaches for discounting actuarial functions. ASTIN Bulletin, 242: ] Parker, G A portfolio of endowent policies and its liiting distribution. ASTIN Bulletin, 261: ] Parker, G Stochastic analysis of the interaction between investent and insurance risks. North Aerican Actuarial Journal, 12: ] Parker, G Stochastic interest rates with actuarial applications. Invited paper in Journal of Applied Stochastic Models and Data Analysis, 14: ] Ross, S. M Introduction to Probability Models. 8th edn. Acadeic Press, San Diego. 18

19 A Proof of Result 7.1 In the proof, we use the forula for coputing expectation by conditioning and the conditional variance forula e.g., see Equation 3.3 p.106 and Proposition 3.1 p.118 in Ross VarS acct r ] = Var δr ES acct = Var δr E Lr ES acct r L r, δr]] + + E δr Var Lr ESr acct r δr] + E δr VarSr acct δr] ] L r, δr] + E Lr VarS acct r ] = Var δr E Lr ERG r L r, δr] EP L r L r, δr] + + E δr Var Lr E RG r EP L r L r, δr] L r, δr + + E δr E Lr Var RG r EP L r L r, δr] L r, δr = Var δr ERG r δr] EP L r δr] + ] L r, δr] + E δr Var Lr ERGr L r, δr] + Var Lr EP Lr L r, δr] 2Cov Lr ERGr L r, δr], EP L r L r, δr] ] + + E δr E Lr Var RG r L r, δr ] = Var δr ERG r δr] + E δr Var Lr EP Lr L r, δr] + + E δr E Lr Var RG r L r, δr ] A.1 + Var δr EP L r δr] + E δr Var Lr EP Lr L r, δr] A.2 2 Cov δr ERG r δr], EP L r δr] + + E δr Cov Lr ERGr L r, δr], EP L r L r, δr]. A.3 Next, we siplify expressions A.1, A.2 and A.3. Applying the conditional variance forula twice, Expression A.1 becoes Var δr ERG r δr] + E δr Var Lr ERGr L r, δr] + E Lr Var RG r L r, δr ] = Var δr ERG r δr] + E δr VarRG r δr] = VarRG r ]. A.4 19

20 To nuerically evaluate Expression A.2 it can be rewritten as Var δr EP L r δr] + E δr Var Lr EP L r L r, δr] = Var δr EP Cj r ] Ee Ir,r+j δr] + j=0 + E δr Var Lr EP Cj r L r ] Ee Ir,r+j δr] j=0 = EP Ci r ] EP Cj r ] Cov δr Ee Ir,r+i δr], Ee Ir,r+j δr] + i=0 j=0 + Cov Lr EP Ci r L r ], EP Cj r L r ] i=0 j=0 i=0 j=0 E δr Ee Ir,r+i δr] Ee Ir,r+j δr] = E Lr EP Ci r L r ] EP Cj r L r ] E δr Ee Ir,r+i δr] Ee Ir,r+j δr] EP Ci r ] EP Cj r ] Ee Ir,r+i ] Ee Ir,r+j ] i=0 j=0 = E Lr EP Ci r L r ] EP Cj r L r ] i=0 j=0 E δr Ee Ir,r+i δr] Ee Ir,r+j δr] 2. EP L r ] A.5 Finally, Expression A.3 siplifies to E δr ERG r δr] EP L r δr] ERG r ] EP L r ] + + E δr E Lr ERG r L r, δr] EP L r L r, δr] ] E δr ERG r δr] EP L r δr] = = = r E Lr ERC j r L r ] EP Ci r L r ] E δr Ee Ij,r δr] Ee Ir,r+i δr] j=0 i=0 ERG r ] EP L r ] r E Lr ERC j r P Ci r L r ] j=0 i=0 E δr Ee Ij,r Ir,r+i δr] ERG r ] EP L r ] r ERCj r P Ci r ] Ee Ij,r Ir,r+i ] ERG r ] EP L r ] j=0 i=0 = ERG r P L r ] ERG r ] EP L r ] = CovRG r, P L r. A.6 20

21 In the above derivation we use the fact that {RCj r L r } and {P Ci r L r } are uncorrelated as well as {e Ij,r δr} and {e Ir,r+i δr}, which can be shown as follows: E e Ij,r e Ir,r+i δr ] = E Ij,r E e Ij,r e Ir,r+i Ij, r, δr ]] Cov e Ij,r, e Ir,r+i δr = 0. Siilarly, E RCj r P Ci r ] L r Cov RCj r, P Cr i L r = 0. = E Ij,r e Ij,r E e Ir,r+i Ij, r, δr ]] = E Ij,r e Ij,r E e Ir,r+i δr ] ] Markovian property = E e Ij,r δr ] E e Ir,r+i δr ], = E RC r j E RCj r P Ci r RC j r ] ], L r = E RC r j RC j r E P Ci r RC j r ] ], L r = E RC r j RC j r E P Ci r ] ] Lr Markovian property = E RCj r ] ] L r E P C r i L r, Now, replacing A.1-A.3 with A.4-A.6 proves the result. 21

22 B Mortality Table x q x x q x x q x x q x